Estimating weights in the synthetic control method involves an optimization procedure that simultaneously selects and aligns control units in order to closely match the treated unit. However, this simultaneous selection and alignment of control units may lead to a loss of efficiency in the synthetic control method. Another concern arising from the aforementioned procedure is its susceptibility to under-fitting due to imperfect pretreatment fit. It is not uncommon for the linear combination, using nonnegative weights, of pre-treatment period outcomes for the control units to inadequately approximate the pre-treatment outcomes for the treated unit. To address both of these issues, this paper proposes a simple and effective method called Synthetic Matching Control (SMC). The SMC method begins by performing the univariate linear regression to establish a proper match between the pre-treatment periods of the control units and the treated unit. Subsequently, a SMC estimator is obtained by synthesizing (taking a weighted average) the matched controls. To determine the weights in the synthesis procedure, we propose an approach that utilizes a criterion of unbiased risk estimator. Theoretically, we show that the synthesis way is asymptotically optimal in the sense of achieving the lowest possible squared error. Extensive numerical experiments highlight the advantages of the SMC method.

翻译：在合成控制方法中估计权重涉及一个同时选择和匹配控制单元的优化过程，以紧密匹配处理单元。然而，这种同时选择和匹配控制单元的方式可能导致合成控制方法效率降低。上述过程引发的另一个问题是，由于预处理期拟合不完美，该方法容易产生欠拟合。控制单元在预处理期结果的带非负权重的线性组合，往往难以充分近似处理单元的预处理期结果。为解决这两个问题，本文提出一种简单有效的方法——合成匹配控制（SMC）。SMC方法首先通过进行单变量线性回归，在控制单元与处理单元的预处理期之间建立适当的匹配。随后，通过对匹配后的控制单元进行合成（采用加权平均）得到SMC估计量。为确定合成过程中的权重，我们提出一种基于无偏风险估计准则的方法。理论上，我们证明该合成方式在达到最低平方误差的意义上是渐近最优的。大量数值实验凸显了SMC方法的优势。